Denote by \(\Omega :=\left\lbrace  x \in \operatorname{aff} \,( \lbrace z_{0}, z_{1}, \ldots , z_{m} \rbrace  ) ~:~ (\forall i \in I) ~ \langle x , z_{i} -z_{0}\rangle  = \lambda _{i} \right\rbrace  \) .
According to the assumption and [1]},
\(z_{1}-z_{0}, \ldots , z_{m}-z_{0} \)  are linearly independent.
Then by [1]}, the Gram matrix \(G(z_{1}-z_{0},\ldots , z_{m}-z_{0})\)  is invertible. Set
\(\begin{pmatrix}\alpha _{1} \\\vdots \\\alpha _{m} \\\end{pmatrix}:= G(z_{1}-z_{0}, \ldots , z_{m}-z_{0})^{-1}\begin{pmatrix}\lambda _{1}-\langle z_{0},z_{1}-z_{0}\rangle  \\\vdots \\\lambda _{m}-\langle z_{0},z_{m}-z_{0}\rangle  \\\end{pmatrix}.\) 
